1Classical probability
IA Probability
1.1 Classical probability
Definition (Classical probability). Classical probability applies in a situation
when there are a finite number of equally likely outcome.
A classical example is the problem of points.
Example. A and B play a game in which they keep throwing coins. If a head
lands, then
A
gets a point. Otherwise,
B
gets a point. The first person to get
10 points wins a prize.
Now suppose
A
has got 8 points and
B
has got 7, but the game has to end
because an earthquake struck. How should they divide the prize? We answer
this by finding the probability of
A
winning. Someone must have won by the
end of 19 rounds, i.e. after 4 more rounds. If
A
wins at least 2 of them, then
A
wins. Otherwise, B wins.
The number of ways this can happen is
4
2
+
4
3
+
4
4
= 11, while there are
16 possible outcomes in total. So A should get 11/16 of the prize.
In general, consider an experiment that has a random outcome.
Definition (Sample space). The set of all possible outcomes is the sample space,
Ω. We can lists the outcomes as ω
1
, ω
2
, ··· ∈ Ω. Each ω ∈ Ω is an outcome.
Definition (Event). A subset of Ω is called an event.
Example. When rolling a dice, the sample space is
{
1
,
2
,
3
,
4
,
5
,
6
}
, and each
item is an outcome. “Getting an odd number” and “getting 3” are two possible
events.
In probability, we will be dealing with sets a lot, so it would be helpful to
come up with some notation.
Definition (Set notations). Given any two events A, B ⊆ Ω,
– The complement of A is A
C
= A
′
=
¯
A = Ω \ A.
– “A or B” is the set A ∪ B.
– “A and B” is the set A ∩ B.
– A and B are mutually exclusive or disjoint if A ∩ B = ∅.
– If A ⊆ B, then A occurring implies B occurring.
Definition (Probability). Suppose Ω =
{ω
1
, ω
2
, ··· , ω
N
}
. Let
A ⊆
Ω be an
event. Then the probability of A is
P(A) =
Number of outcomes in A
Number of outcomes in Ω
=
|A|
N
.
Here we are assuming that each outcome is equally likely to happen, which
is the case in (fair) dice rolls and coin flips.
Example. Suppose
r
digits are drawn at random from a table of random digits
from 0 to 9. What is the probability that
(i) No digit exceeds k;
(ii) The largest digit drawn is k?
The sample space is Ω = {(a
1
, a
2
, ··· , a
r
) : 0 ≤ a
i
≤ 9}. Then |Ω| = 10
r
.
Let
A
k
= [
no digit exceeds k
] =
{
(
a
1
, ··· , a
r
) : 0
≤ a
i
≤ k}
. Then
|A
k
|
=
(k + 1)
r
. So
P (A
k
) =
(k + 1)
r
10
r
.
Now let
B
k
= [
largest digit drawn is k
]. We can find this by finding all outcomes
in which no digits exceed
k
, and subtract it by the number of outcomes in which
no digit exceeds k − 1. So |B
k
| = |A
k
| − |A
k−1
| and
P (B
k
) =
(k + 1)
r
− k
r
10
r
.